23 Trigonometric identities and functions; Question 23.21

[Eigenvalue]

23.21 Verify the following are identities:

a) [tex]\frac{1}{sin(x)}[/tex]-sin(x)=[tex]\frac{cos²(x)}{sin(x)}[/tex]

b) [tex]\frac{1+sin(x)}{cos(x)}[/tex]+[tex]\frac{cos(x)}{1+sin(x)}[/tex]=2sec(x)

c) (sec(x)+tan(x))²=[tex]\frac{1+sin(x)}{1-sin(x)}[/tex]

[Eigenvalue]

a) [tex]\frac{1}{sin(x)}[/tex]-sin(x)=[tex]\frac{cos²(x)}{sin(x)}[/tex]

[tex]\frac{1}{sin(x)}[/tex]-sin(x)

=[tex]\frac{1-sin²(x)}{sin(x)}[/tex]

=[tex]\frac{cos²(x)}{sin(x)}[/tex]

b) [tex]\frac{1+sin(x)}{cos(x)}[/tex]+[tex]\frac{cos(x)}{1+sin(x)}[/tex]=2sec(x)

[tex]\frac{1+sin(x)}{cos(x)}[/tex]+[tex]\frac{cos(x)}{1+sin(x)}[/tex]

=[tex]\frac{(1+sin(x))²+cos²(x)}{cos(x)(1+sin(x)}[/tex]

=[tex]\frac{1+2sin(x)+sin²(x)+cos²(x)}{cos(x)(1+sin(x))}[/tex]

=[tex]\frac{2+2sin(x)}{cos(x)(1+sin(x))}[/tex]

=[tex]\frac{2}{cos(x)}[/tex]

=sec(x)

c) (sec(x)+tan(x))²=[tex]\frac{1+sin(x)}{1-sin(x)}[/tex]
(sec(x)+tan(x))²

=sec²(x)+tan²(x)+2sec(x)tan(x)

=[tex]\frac{1}{cos²(x)}[/tex]+[tex]\frac{sin²(x)}{cos²(x)}[/tex]+2([tex]\frac{1}{cos(x)}[/tex]*[tex]\frac{sin(x)}{cos(x)}[/tex]

=[tex]\frac{1+2sin(x)+sin²(x)}{cos²(x)}[/tex]

=[tex]\frac{(1+sin(x))²}{1-sin²(x)}[/tex]

=[tex]\frac{1+sin(x)}{1-sin(x)}[/tex]